Problem: Multiply and simplify the following complex numbers: $({5-5i}) \cdot ({-3+5i})$
Answer: Complex numbers are multiplied like any two binomials. First use the distributive property: $ ({5-5i}) \cdot ({-3+5i}) = $ $ ({5} \cdot {-3}) + ({5} \cdot {5i}) + ({-5i} \cdot {-3}) + ({-5i} \cdot {5i}) $ Then simplify the terms: $ (-15) + (25i) + (15i) + (-25i^2) $ Imaginary unit multiples can be grouped together. $ -15 + (25 + 15)i - 25 i^2 $ After we plug in $i^2 = -1$, the result becomes $ -15 + (25 + 15)i - (-25) $ The result is simplified: $ (-15 + 25) + (40i) = 10+40i $